目录
常见普通型生成函数($OGF$)
形如$F(x)=sum_{i=0}^{infty}f_ix^i$：
$$egin{align*}
&<1,0,0,cdots>[i==0]&1\
&<1,1,1,cdots>1&frac{1}{1-x}\
&<1,2,3,cdots>i&frac{1}{(1-x)^2}\
&<1,-1,1,-1,cdots>(-1)^i&frac{1}{1-x}\
&<0,1,frac{1}{2},frac{1}{3},cdots>frac{1}{i}(0<i)&-ln(1-x)\
&<1,1,frac{1}{2},frac{1}{6},frac{1}{24},cdots>frac{1}{i!}&e^x\
&<1,a,a^2,a^3,cdots>a^i&frac{1}{1-ax}\
&<inom{n}{0},inom{n}{1},inom{n}{2},cdots>inom{n}{i}&(1+x)^n\
&<inom{n-1}{0},inom{n}{1},inom{n+1}{2},cdots>inom{n-i+1}{i}&frac{1}{(1-x)^n}\
&<0,a,frac{a^2}{2},frac{a^3}{3},cdots>frac{a^i}{i}(0<i)&-ln(1-ax)\
end{align*}$$
常见指数型生成函数($EGF$)
形如$F(x)=sum_{i=0}^{infty}frac{f_i}{i!}x^i$：
$$egin{align*}
&<1,1,1,cdots>1&e^x\
&<0,1,2,3,cdots>i&xe^x\
&<1,a,a^2,a^3,cdots>a^i&e^{ax}\
&<1,a,a^{underline{2}},a^{underline{3}},cdots>a^{underline{i}}&(1+x)^a\
&<0,1,0,-1,0,1,0,-1,cdots>[2
mid i](-1)^{frac{i-1}{2}}&sin(x)\
&<1,0,-1,0,1,0,-1,0,cdots>[2|i](-1)^{frac{i}{2}}&cos(x)\
end{align*}$$
自然数幂和
求数列$k$次方和
给定数列$a$，对于任一$1 leqslant k leqslant m$求$sum_{i=1}^na_i^k$
考虑其生成函数，则有：$$egin{align*}F(x)&=sum_{j=0}^{infty}sum_{i=1}^na_i^jx^j\&=sum_{i=1}^nsum_{j=0}^{infty}(a_ix)^j\&=sum_{i=1}^nfrac{1}{1-a_ix}\&=sum_{i=1}^n1+frac{a_ix}{1-a_ix}\&=n-xsum_{i=1}^nfrac{-a_ix}{1-a_ix}\&=n-xsum_{i=1}^nln'(1-a_ix)\&=n-xln'(prod_{i=1}^n(1-a_ix))end{align*}$$
$prod_{i=1}^n(1-a_ix)$分治FFT即可
$O(Nlog^2N)$